#984 - Knights and Horses
In Western chess, a knight is a piece that moves either two squares horizontally and one square vertically, or one square horizontally and two squares vertically, and is capable of jumping over any intervening pieces.
Chinese chess has a similar piece called the horse, whose moves have an identical displacement as a knight's move; however, a horse, unlike a knight, is unable to jump over intervening pieces.
More specifically, a horse's move consists of two steps: An orthogonal move of one square, followed by a diagonal move by one square in the same direction as the orthogonal move. If the orthogonal square is occupied by another piece, the horse is unable to move in that direction.
Specifically the horse in the centre of the above board can move to the squares \(b_{11},b_{12},b_{21},b_{22},b_{31},b_{32},b_{41},b_{42}\) providing the squares \(a_{1},a_{2},a_{3},a_{4}\) are unoccupied. For example, if \(a_2\) was occupied then it could not move to \(b_{21}\) or \(b_{22}\).
A set of squares on a chessboard is called knight-connected if a knight can travel between any two squares in the set using only legal moves without using any squares not in the set.
A set of squares on a chessboard is called horse-disjoint if, when a horse is placed on every square in the set (and no other square), no horse can attack any other.
Let \(f(N)\) be the number of knight-connected, horse-disjoint non-empty subsets of an \(N\times N\) chessboard. For example, \(f(3) = 9\), consisting of the nine singleton sets. You are also given that \(f(5) = 903, f(100) = 8658918531876\), and \(f(10000) \equiv 377956308 \bmod 10^9+7\).
Find \(f(10^{18})\). Give your answer modulo \(10^9+7\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=984. Published Saturday, 14th February 2026, 07:00 pm. Solved by 103 members at time of mirroring.
Why this is useful
Algorithmic Development. Optimal substructure and state-space reasoning are exactly how American-option pricing and optimal execution are solved (Phases 13, 16).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.8 Bit Manipulation and State Compression · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.11 Integer Partitions and Counting Structures · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 19.2 Primes, Sieves, and Integer Factorization · 3.1 Vectors, Multivariable Functions, and Level Sets · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #420 · #833 · #476
Concepts: combinatorics geometry graph-theory number-theory brute-force-reduction
Likely techniques: bfs-dfs bitmask-dp hashing modular-exponentiation
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 984? Is it a bound (10^18), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single exact integer. Required format: Give your answer modulo 10^9+7.
- Write out, in your own words, the definition of horse-disjoint if as the statement gives it. Which integers/objects are excluded by that definition?
- What do the arguments of f(N), f(3) mean, and what is the value's type (count, sum, probability)?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10^18?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10^18 and the cost of testing one.
- Which geometry fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says geometry / bitmask-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10^18, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it (the statement says: "For example, if a_2 was occupied then it could not move to b_21 or b_22.") - then run it. A surprise here is worth more than an hour of debugging later.
Scratchpad
Mathematical notes, formulas, pseudocode, hypotheses, complexity notes. Saved automatically with your progress.
Python workbench
Real Python (Pyodide) in a sandboxed Web Worker - no network, no filesystem, no DOM access. Ctrl/Cmd+Enter runs. Escape leaves the editor. Stop terminates the worker.
Check your answer
Answers are checked against a salted hash held in a separate file - not printed in this page. This prevents accidental spoilers; it is not cryptographic protection (see the build notes).
Progressive hints
Optimization
You have a correct answer. That is the start of the learning, not the end.
- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
- Reduce memory. Can you stream, or keep only the last k states?
- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
- Prove the optimized version computes the same thing.
- Compare two implementations and time them.
Explain it
Which step of your solution were you least confident about, and what evidence would settle it?
What did you try first, and what specifically made you abandon it - a proof, a timing, or a wrong small-case answer?
Where did the geometry structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the bitmask-dp idea faster? Which words in the statement were pointing at it, and did you notice them?
What was the bug that cost you the most time, and what CLASS of bug was it (off-by-one, definition misread, precision, state under-specified)?
How would your solution change if the bound 10^18 were multiplied by 1000? Does it survive, or does it need a different idea?
What is the honest complexity of what you wrote (not what you intended), and where is the remaining slack?
Which problem you have already solved is this most similar to, and what is the shared skeleton - is it really 'bitmask-dp' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
Self-assess (mastery is not a correct number)
You reach Mastered only when you have solved it, rated yourself at least Solid across the dimensions, and written a real explanation.
Confidence
Low confidence schedules this problem for spaced review, even if you solved it.
Mastery check
- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: #420 · #833 · #476
- Transfer: where else does this technique appear? Name a lesson and a real computational setting.
- Spaced re-attempt: come back after the review interval and re-solve it with no hints.